Mixing
Theorem 3.7 in Baby Rudin: The subsequential limits of a sequence in a metric space form a closed set
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Here's Theorem 3.7 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition.
The subsequential limits of a sequence $(p_n)$ in a metric space $X$ form a closed subset of $X$.
And, here's Rudin's proof.
Let $E^*$ be the set of all subsequential limits of $(p_n)$ and let $q$ be a limit point of $E^*$. We have to show that $q in E^*$.
Choose $n_1$ so that $p_{n_1} neq q$. (If no such $n_1$ exists, then $E^*$ has only one point, and there is nothing to prove. ) Put $delta = d(q, p_{n_1})$. Suppose $n_1, ldots, n_{i-1}$ are chosen. Since $q$ is a limit point of $E^*$, there is an $x in E^*$ with $d(x, q) < 2^{-i} delta$. Since $x in E^*$, there is an $n_i > n_{i-1}$ such that $d(x, p_{n_i}) < 2^{-i} delta$. Thus $$d(q, p_{n_i}) leq 2^{1-i} delta$$ for $i = 1, 2, 3, ldots$. This says that $(p_{n_i})$ converges to $q$. Hence $q in E^*$.
Now here's my reading of Rudin's proof.
If the set $E^*$ of all the subsequential limits of the sequence $(p_n)$ has no limit points, then the set of all the limit points of the set $E^*$ is empty and is therefore contained in $E^*$.
So let's suppose that $q$ is a limit point of the set $E^*$. If $p_n = q$ for all $n in mathbb{N}$, then the sequence $(p_n)$, being a constant sequence, converges to $q$, and so every subsequence of $(p_n)$ also converges to $q$; therefore the set $E^*$ consists of a single point $q$ and thus cannot have limit points. So there is a natural number $n$ for which $p_n neq q$. Let $n_1$ be the smallest such natural number.
Let's put $delta = dleft(q, p_{n_1}right)$. Then $delta > 0$. Now since $q$ is a limit point of the set $E^*$, there exists a point $x_1 in E^*$ such that $$0 < dleft(q, x_1right) < frac{delta}{4}.$$
Now since $x_1$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_1 colon mathbb{N} to mathbb{N}$ such that $$x_1 = lim_{n to infty} p_{varphi_1(n)}.$$ So there exists a natural number $N_1$ such that $$dleft( x_1 , p_{varphi_1(n)} right) < frac{delta}{4} $$ for all natural numbers $n$ such that $n > N_1$.
We note that, for each $n in mathbb{N}$, the inequality $n leq varphi_1(n)$ holds.
Let $n_2$ be the natural number defined as $$n_2 colon= max left( varphi_1(n_1 + 1) , varphi_1(N_1 + 1) right).$$ Then $n_2 > N_1$ and so $$ dleft( q , p_{n_2} right) leq dleft( q, x_1 right) + dleft( x_1 , p_{n_2} right) < frac{delta}{4} + frac{delta}{4} = frac{delta}{2}.$$
Now as $q$ is a limit point of $E^*$, there exists a point $x_2 in E^*$ such that $$0 < d(q, x_2) < frac{ delta}{8}.$$
Moreover, since $x_2$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_2 colon mathbb{N} to mathbb{N}$ such that $$x_2 = lim_{n to infty} p_{varphi_2(n)}.$$ So there is a natural number $N_2$ such that $$ dleft( x_2 , p_{varphi_2(n)} right) < frac{delta}{8}$$ for all natural numbers $n > N_2$.
Note that, for each $n in mathbb{N}$, we have $n leq varphi_2 (n)$.
Now let $$n_3 colon= max left( varphi_2(n_2 + 1) , varphi_2 ( N_2 + 1) right).$$ Then $n_3$ is a natural number greater than $N_2$ and so we have $$dleft( q , p_{n_3} right) leq dleft( q , x_2 right) + dleft( x_2 , p_{n_3} right) < frac{delta}{8} + frac{delta}{8} = frac{delta}{4}.$$
We note that $$n_3 geq varphi_2(n_2 + 1) > varphi_2(n_2) geq n_2 geq varphi_1 (n_1 + 1) > varphi_1 (n_1) geq n_1.$$ That is, $n_1, n_2, n_3$ are all natural numbers such that $$n_1 < n_2 < n_3.$$
Continuing in this way, we obtain a subsequence $(p_{n_i})$, where $n_1, n_2, n_3, ldots in mathbb{N}$ and $n_1 < n_2 < n_3 < cdots$, such that $$dleft( q , p_{n_i} right) leq frac{2delta}{2^i}$$ for all $i in mathbb{N}$.
We now show that this subsequence $(p_{n_i})$ converges to $q$. Let $varepsilon > 0$ be given. Then we can find a natural number $$K > frac{2delta}{varepsilon}.$$ Then $$2^K > K > frac{2delta}{varepsilon}.$$ So for any natural number $i > K$, we have $$2^i > 2^K > frac{2delta}{varepsilon}$$ and so $$ dleft( q , p_{n_i} right) leq frac{2 delta}{2^i} < varepsilon. $$
Thus every limit point $q$ of the set $E^*$ is also an element of $E^*$. Hence $E^*$ is closed.
Is my reading of Rudin's proof correct? If not, what is it that I'm missing?
real-analysis sequences-and-series analysis proof-verification metric-spaces
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
$endgroup$
|
show 3 more comments
$begingroup$
Here's Theorem 3.7 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition.
The subsequential limits of a sequence $(p_n)$ in a metric space $X$ form a closed subset of $X$.
And, here's Rudin's proof.
Let $E^*$ be the set of all subsequential limits of $(p_n)$ and let $q$ be a limit point of $E^*$. We have to show that $q in E^*$.
Choose $n_1$ so that $p_{n_1} neq q$. (If no such $n_1$ exists, then $E^*$ has only one point, and there is nothing to prove. ) Put $delta = d(q, p_{n_1})$. Suppose $n_1, ldots, n_{i-1}$ are chosen. Since $q$ is a limit point of $E^*$, there is an $x in E^*$ with $d(x, q) < 2^{-i} delta$. Since $x in E^*$, there is an $n_i > n_{i-1}$ such that $d(x, p_{n_i}) < 2^{-i} delta$. Thus $$d(q, p_{n_i}) leq 2^{1-i} delta$$ for $i = 1, 2, 3, ldots$. This says that $(p_{n_i})$ converges to $q$. Hence $q in E^*$.
Now here's my reading of Rudin's proof.
If the set $E^*$ of all the subsequential limits of the sequence $(p_n)$ has no limit points, then the set of all the limit points of the set $E^*$ is empty and is therefore contained in $E^*$.
So let's suppose that $q$ is a limit point of the set $E^*$. If $p_n = q$ for all $n in mathbb{N}$, then the sequence $(p_n)$, being a constant sequence, converges to $q$, and so every subsequence of $(p_n)$ also converges to $q$; therefore the set $E^*$ consists of a single point $q$ and thus cannot have limit points. So there is a natural number $n$ for which $p_n neq q$. Let $n_1$ be the smallest such natural number.
Let's put $delta = dleft(q, p_{n_1}right)$. Then $delta > 0$. Now since $q$ is a limit point of the set $E^*$, there exists a point $x_1 in E^*$ such that $$0 < dleft(q, x_1right) < frac{delta}{4}.$$
Now since $x_1$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_1 colon mathbb{N} to mathbb{N}$ such that $$x_1 = lim_{n to infty} p_{varphi_1(n)}.$$ So there exists a natural number $N_1$ such that $$dleft( x_1 , p_{varphi_1(n)} right) < frac{delta}{4} $$ for all natural numbers $n$ such that $n > N_1$.
We note that, for each $n in mathbb{N}$, the inequality $n leq varphi_1(n)$ holds.
Let $n_2$ be the natural number defined as $$n_2 colon= max left( varphi_1(n_1 + 1) , varphi_1(N_1 + 1) right).$$ Then $n_2 > N_1$ and so $$ dleft( q , p_{n_2} right) leq dleft( q, x_1 right) + dleft( x_1 , p_{n_2} right) < frac{delta}{4} + frac{delta}{4} = frac{delta}{2}.$$
Now as $q$ is a limit point of $E^*$, there exists a point $x_2 in E^*$ such that $$0 < d(q, x_2) < frac{ delta}{8}.$$
Moreover, since $x_2$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_2 colon mathbb{N} to mathbb{N}$ such that $$x_2 = lim_{n to infty} p_{varphi_2(n)}.$$ So there is a natural number $N_2$ such that $$ dleft( x_2 , p_{varphi_2(n)} right) < frac{delta}{8}$$ for all natural numbers $n > N_2$.
Note that, for each $n in mathbb{N}$, we have $n leq varphi_2 (n)$.
Now let $$n_3 colon= max left( varphi_2(n_2 + 1) , varphi_2 ( N_2 + 1) right).$$ Then $n_3$ is a natural number greater than $N_2$ and so we have $$dleft( q , p_{n_3} right) leq dleft( q , x_2 right) + dleft( x_2 , p_{n_3} right) < frac{delta}{8} + frac{delta}{8} = frac{delta}{4}.$$
We note that $$n_3 geq varphi_2(n_2 + 1) > varphi_2(n_2) geq n_2 geq varphi_1 (n_1 + 1) > varphi_1 (n_1) geq n_1.$$ That is, $n_1, n_2, n_3$ are all natural numbers such that $$n_1 < n_2 < n_3.$$
Continuing in this way, we obtain a subsequence $(p_{n_i})$, where $n_1, n_2, n_3, ldots in mathbb{N}$ and $n_1 < n_2 < n_3 < cdots$, such that $$dleft( q , p_{n_i} right) leq frac{2delta}{2^i}$$ for all $i in mathbb{N}$.
We now show that this subsequence $(p_{n_i})$ converges to $q$. Let $varepsilon > 0$ be given. Then we can find a natural number $$K > frac{2delta}{varepsilon}.$$ Then $$2^K > K > frac{2delta}{varepsilon}.$$ So for any natural number $i > K$, we have $$2^i > 2^K > frac{2delta}{varepsilon}$$ and so $$ dleft( q , p_{n_i} right) leq frac{2 delta}{2^i} < varepsilon. $$
Thus every limit point $q$ of the set $E^*$ is also an element of $E^*$. Hence $E^*$ is closed.
Is my reading of Rudin's proof correct? If not, what is it that I'm missing?
real-analysis sequences-and-series analysis proof-verification metric-spaces
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
$endgroup$
$begingroup$
This is a rather long restatement of Rudin's proof. I'm sure there are people (myself included) who will be happy to help but don't necessarily want to proofread the whole thing. Can you highlight the part(s) that you are not sure about?
$endgroup$
– Bungo
Apr 19 '16 at 19:08
1
$begingroup$
@Bungo please have a look at my post after I've edited it to remove some errors. Do you really think my rendering of Rudin's proof is too lengthy for other people to be able to take time reading it? Actually, this proof has been proving to be a bit too hard for me to imbibe for such a long time. So I thought I should try to write it out the way I've understood it and ask for the Maths SE community's opinion as to my understanding and presentation of the proof.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:15
2
$begingroup$
In general, a shorter more targeted question will attract more readers. I think including the full proof is OK, but it would be helpful if you can point out any part(s) that you would like us to focus on. Also, may I suggest adding the "proof-verification" tag.
$endgroup$
– Bungo
Apr 19 '16 at 19:23
$begingroup$
@Bungo what I'm specifically interested to know is whether or not I've read Rudin's inductive argument correctly, and whether or not I've managed to present it in a more easily understandable way.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:43
$begingroup$
@Bungo please have a look at my post now.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:44
|
show 3 more comments
$begingroup$
Here's Theorem 3.7 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition.
The subsequential limits of a sequence $(p_n)$ in a metric space $X$ form a closed subset of $X$.
And, here's Rudin's proof.
Let $E^*$ be the set of all subsequential limits of $(p_n)$ and let $q$ be a limit point of $E^*$. We have to show that $q in E^*$.
Choose $n_1$ so that $p_{n_1} neq q$. (If no such $n_1$ exists, then $E^*$ has only one point, and there is nothing to prove. ) Put $delta = d(q, p_{n_1})$. Suppose $n_1, ldots, n_{i-1}$ are chosen. Since $q$ is a limit point of $E^*$, there is an $x in E^*$ with $d(x, q) < 2^{-i} delta$. Since $x in E^*$, there is an $n_i > n_{i-1}$ such that $d(x, p_{n_i}) < 2^{-i} delta$. Thus $$d(q, p_{n_i}) leq 2^{1-i} delta$$ for $i = 1, 2, 3, ldots$. This says that $(p_{n_i})$ converges to $q$. Hence $q in E^*$.
Now here's my reading of Rudin's proof.
If the set $E^*$ of all the subsequential limits of the sequence $(p_n)$ has no limit points, then the set of all the limit points of the set $E^*$ is empty and is therefore contained in $E^*$.
So let's suppose that $q$ is a limit point of the set $E^*$. If $p_n = q$ for all $n in mathbb{N}$, then the sequence $(p_n)$, being a constant sequence, converges to $q$, and so every subsequence of $(p_n)$ also converges to $q$; therefore the set $E^*$ consists of a single point $q$ and thus cannot have limit points. So there is a natural number $n$ for which $p_n neq q$. Let $n_1$ be the smallest such natural number.
Let's put $delta = dleft(q, p_{n_1}right)$. Then $delta > 0$. Now since $q$ is a limit point of the set $E^*$, there exists a point $x_1 in E^*$ such that $$0 < dleft(q, x_1right) < frac{delta}{4}.$$
Now since $x_1$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_1 colon mathbb{N} to mathbb{N}$ such that $$x_1 = lim_{n to infty} p_{varphi_1(n)}.$$ So there exists a natural number $N_1$ such that $$dleft( x_1 , p_{varphi_1(n)} right) < frac{delta}{4} $$ for all natural numbers $n$ such that $n > N_1$.
We note that, for each $n in mathbb{N}$, the inequality $n leq varphi_1(n)$ holds.
Let $n_2$ be the natural number defined as $$n_2 colon= max left( varphi_1(n_1 + 1) , varphi_1(N_1 + 1) right).$$ Then $n_2 > N_1$ and so $$ dleft( q , p_{n_2} right) leq dleft( q, x_1 right) + dleft( x_1 , p_{n_2} right) < frac{delta}{4} + frac{delta}{4} = frac{delta}{2}.$$
Now as $q$ is a limit point of $E^*$, there exists a point $x_2 in E^*$ such that $$0 < d(q, x_2) < frac{ delta}{8}.$$
Moreover, since $x_2$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_2 colon mathbb{N} to mathbb{N}$ such that $$x_2 = lim_{n to infty} p_{varphi_2(n)}.$$ So there is a natural number $N_2$ such that $$ dleft( x_2 , p_{varphi_2(n)} right) < frac{delta}{8}$$ for all natural numbers $n > N_2$.
Note that, for each $n in mathbb{N}$, we have $n leq varphi_2 (n)$.
Now let $$n_3 colon= max left( varphi_2(n_2 + 1) , varphi_2 ( N_2 + 1) right).$$ Then $n_3$ is a natural number greater than $N_2$ and so we have $$dleft( q , p_{n_3} right) leq dleft( q , x_2 right) + dleft( x_2 , p_{n_3} right) < frac{delta}{8} + frac{delta}{8} = frac{delta}{4}.$$
We note that $$n_3 geq varphi_2(n_2 + 1) > varphi_2(n_2) geq n_2 geq varphi_1 (n_1 + 1) > varphi_1 (n_1) geq n_1.$$ That is, $n_1, n_2, n_3$ are all natural numbers such that $$n_1 < n_2 < n_3.$$
Continuing in this way, we obtain a subsequence $(p_{n_i})$, where $n_1, n_2, n_3, ldots in mathbb{N}$ and $n_1 < n_2 < n_3 < cdots$, such that $$dleft( q , p_{n_i} right) leq frac{2delta}{2^i}$$ for all $i in mathbb{N}$.
We now show that this subsequence $(p_{n_i})$ converges to $q$. Let $varepsilon > 0$ be given. Then we can find a natural number $$K > frac{2delta}{varepsilon}.$$ Then $$2^K > K > frac{2delta}{varepsilon}.$$ So for any natural number $i > K$, we have $$2^i > 2^K > frac{2delta}{varepsilon}$$ and so $$ dleft( q , p_{n_i} right) leq frac{2 delta}{2^i} < varepsilon. $$
Thus every limit point $q$ of the set $E^*$ is also an element of $E^*$. Hence $E^*$ is closed.
Is my reading of Rudin's proof correct? If not, what is it that I'm missing?
real-analysis sequences-and-series analysis proof-verification metric-spaces
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
$endgroup$
Here's Theorem 3.7 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition.
The subsequential limits of a sequence $(p_n)$ in a metric space $X$ form a closed subset of $X$.
And, here's Rudin's proof.
Let $E^*$ be the set of all subsequential limits of $(p_n)$ and let $q$ be a limit point of $E^*$. We have to show that $q in E^*$.
Choose $n_1$ so that $p_{n_1} neq q$. (If no such $n_1$ exists, then $E^*$ has only one point, and there is nothing to prove. ) Put $delta = d(q, p_{n_1})$. Suppose $n_1, ldots, n_{i-1}$ are chosen. Since $q$ is a limit point of $E^*$, there is an $x in E^*$ with $d(x, q) < 2^{-i} delta$. Since $x in E^*$, there is an $n_i > n_{i-1}$ such that $d(x, p_{n_i}) < 2^{-i} delta$. Thus $$d(q, p_{n_i}) leq 2^{1-i} delta$$ for $i = 1, 2, 3, ldots$. This says that $(p_{n_i})$ converges to $q$. Hence $q in E^*$.
Now here's my reading of Rudin's proof.
If the set $E^*$ of all the subsequential limits of the sequence $(p_n)$ has no limit points, then the set of all the limit points of the set $E^*$ is empty and is therefore contained in $E^*$.
So let's suppose that $q$ is a limit point of the set $E^*$. If $p_n = q$ for all $n in mathbb{N}$, then the sequence $(p_n)$, being a constant sequence, converges to $q$, and so every subsequence of $(p_n)$ also converges to $q$; therefore the set $E^*$ consists of a single point $q$ and thus cannot have limit points. So there is a natural number $n$ for which $p_n neq q$. Let $n_1$ be the smallest such natural number.
Let's put $delta = dleft(q, p_{n_1}right)$. Then $delta > 0$. Now since $q$ is a limit point of the set $E^*$, there exists a point $x_1 in E^*$ such that $$0 < dleft(q, x_1right) < frac{delta}{4}.$$
Now since $x_1$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_1 colon mathbb{N} to mathbb{N}$ such that $$x_1 = lim_{n to infty} p_{varphi_1(n)}.$$ So there exists a natural number $N_1$ such that $$dleft( x_1 , p_{varphi_1(n)} right) < frac{delta}{4} $$ for all natural numbers $n$ such that $n > N_1$.
We note that, for each $n in mathbb{N}$, the inequality $n leq varphi_1(n)$ holds.
Let $n_2$ be the natural number defined as $$n_2 colon= max left( varphi_1(n_1 + 1) , varphi_1(N_1 + 1) right).$$ Then $n_2 > N_1$ and so $$ dleft( q , p_{n_2} right) leq dleft( q, x_1 right) + dleft( x_1 , p_{n_2} right) < frac{delta}{4} + frac{delta}{4} = frac{delta}{2}.$$
Now as $q$ is a limit point of $E^*$, there exists a point $x_2 in E^*$ such that $$0 < d(q, x_2) < frac{ delta}{8}.$$
Moreover, since $x_2$ is a subsequential limit of the sequence $(p_n)$, there is a strictly increasing function $varphi_2 colon mathbb{N} to mathbb{N}$ such that $$x_2 = lim_{n to infty} p_{varphi_2(n)}.$$ So there is a natural number $N_2$ such that $$ dleft( x_2 , p_{varphi_2(n)} right) < frac{delta}{8}$$ for all natural numbers $n > N_2$.
Note that, for each $n in mathbb{N}$, we have $n leq varphi_2 (n)$.
Now let $$n_3 colon= max left( varphi_2(n_2 + 1) , varphi_2 ( N_2 + 1) right).$$ Then $n_3$ is a natural number greater than $N_2$ and so we have $$dleft( q , p_{n_3} right) leq dleft( q , x_2 right) + dleft( x_2 , p_{n_3} right) < frac{delta}{8} + frac{delta}{8} = frac{delta}{4}.$$
We note that $$n_3 geq varphi_2(n_2 + 1) > varphi_2(n_2) geq n_2 geq varphi_1 (n_1 + 1) > varphi_1 (n_1) geq n_1.$$ That is, $n_1, n_2, n_3$ are all natural numbers such that $$n_1 < n_2 < n_3.$$
Continuing in this way, we obtain a subsequence $(p_{n_i})$, where $n_1, n_2, n_3, ldots in mathbb{N}$ and $n_1 < n_2 < n_3 < cdots$, such that $$dleft( q , p_{n_i} right) leq frac{2delta}{2^i}$$ for all $i in mathbb{N}$.
We now show that this subsequence $(p_{n_i})$ converges to $q$. Let $varepsilon > 0$ be given. Then we can find a natural number $$K > frac{2delta}{varepsilon}.$$ Then $$2^K > K > frac{2delta}{varepsilon}.$$ So for any natural number $i > K$, we have $$2^i > 2^K > frac{2delta}{varepsilon}$$ and so $$ dleft( q , p_{n_i} right) leq frac{2 delta}{2^i} < varepsilon. $$
Thus every limit point $q$ of the set $E^*$ is also an element of $E^*$. Hence $E^*$ is closed.
Is my reading of Rudin's proof correct? If not, what is it that I'm missing?
real-analysis sequences-and-series analysis proof-verification metric-spaces
real-analysis sequences-and-series analysis proof-verification metric-spaces
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
edited Apr 20 '16 at 5:04
Saaqib Mahmood
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
asked Apr 19 '16 at 18:39
Saaqib MahmoodSaaqib Mahmood
7,95942581
7,95942581
$begingroup$
This is a rather long restatement of Rudin's proof. I'm sure there are people (myself included) who will be happy to help but don't necessarily want to proofread the whole thing. Can you highlight the part(s) that you are not sure about?
$endgroup$
– Bungo
Apr 19 '16 at 19:08
1
$begingroup$
@Bungo please have a look at my post after I've edited it to remove some errors. Do you really think my rendering of Rudin's proof is too lengthy for other people to be able to take time reading it? Actually, this proof has been proving to be a bit too hard for me to imbibe for such a long time. So I thought I should try to write it out the way I've understood it and ask for the Maths SE community's opinion as to my understanding and presentation of the proof.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:15
2
$begingroup$
In general, a shorter more targeted question will attract more readers. I think including the full proof is OK, but it would be helpful if you can point out any part(s) that you would like us to focus on. Also, may I suggest adding the "proof-verification" tag.
$endgroup$
– Bungo
Apr 19 '16 at 19:23
$begingroup$
@Bungo what I'm specifically interested to know is whether or not I've read Rudin's inductive argument correctly, and whether or not I've managed to present it in a more easily understandable way.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:43
$begingroup$
@Bungo please have a look at my post now.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:44
|
show 3 more comments
$begingroup$
This is a rather long restatement of Rudin's proof. I'm sure there are people (myself included) who will be happy to help but don't necessarily want to proofread the whole thing. Can you highlight the part(s) that you are not sure about?
$endgroup$
– Bungo
Apr 19 '16 at 19:08
1
$begingroup$
@Bungo please have a look at my post after I've edited it to remove some errors. Do you really think my rendering of Rudin's proof is too lengthy for other people to be able to take time reading it? Actually, this proof has been proving to be a bit too hard for me to imbibe for such a long time. So I thought I should try to write it out the way I've understood it and ask for the Maths SE community's opinion as to my understanding and presentation of the proof.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:15
2
$begingroup$
In general, a shorter more targeted question will attract more readers. I think including the full proof is OK, but it would be helpful if you can point out any part(s) that you would like us to focus on. Also, may I suggest adding the "proof-verification" tag.
$endgroup$
– Bungo
Apr 19 '16 at 19:23
$begingroup$
@Bungo what I'm specifically interested to know is whether or not I've read Rudin's inductive argument correctly, and whether or not I've managed to present it in a more easily understandable way.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:43
$begingroup$
@Bungo please have a look at my post now.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:44
$begingroup$
This is a rather long restatement of Rudin's proof. I'm sure there are people (myself included) who will be happy to help but don't necessarily want to proofread the whole thing. Can you highlight the part(s) that you are not sure about?
$endgroup$
– Bungo
Apr 19 '16 at 19:08
$begingroup$
This is a rather long restatement of Rudin's proof. I'm sure there are people (myself included) who will be happy to help but don't necessarily want to proofread the whole thing. Can you highlight the part(s) that you are not sure about?
$endgroup$
– Bungo
Apr 19 '16 at 19:08
1
1
$begingroup$
@Bungo please have a look at my post after I've edited it to remove some errors. Do you really think my rendering of Rudin's proof is too lengthy for other people to be able to take time reading it? Actually, this proof has been proving to be a bit too hard for me to imbibe for such a long time. So I thought I should try to write it out the way I've understood it and ask for the Maths SE community's opinion as to my understanding and presentation of the proof.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:15
$begingroup$
@Bungo please have a look at my post after I've edited it to remove some errors. Do you really think my rendering of Rudin's proof is too lengthy for other people to be able to take time reading it? Actually, this proof has been proving to be a bit too hard for me to imbibe for such a long time. So I thought I should try to write it out the way I've understood it and ask for the Maths SE community's opinion as to my understanding and presentation of the proof.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:15
2
2
$begingroup$
In general, a shorter more targeted question will attract more readers. I think including the full proof is OK, but it would be helpful if you can point out any part(s) that you would like us to focus on. Also, may I suggest adding the "proof-verification" tag.
$endgroup$
– Bungo
Apr 19 '16 at 19:23
$begingroup$
In general, a shorter more targeted question will attract more readers. I think including the full proof is OK, but it would be helpful if you can point out any part(s) that you would like us to focus on. Also, may I suggest adding the "proof-verification" tag.
$endgroup$
– Bungo
Apr 19 '16 at 19:23
$begingroup$
@Bungo what I'm specifically interested to know is whether or not I've read Rudin's inductive argument correctly, and whether or not I've managed to present it in a more easily understandable way.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:43
$begingroup$
@Bungo what I'm specifically interested to know is whether or not I've read Rudin's inductive argument correctly, and whether or not I've managed to present it in a more easily understandable way.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:43
$begingroup$
@Bungo please have a look at my post now.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:44
$begingroup$
@Bungo please have a look at my post now.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:44
|
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The hard bit in this theorem to show that $E^* subset X$. In his proof Rudin is trying to make a direct proof by complete induction that for any limit point $q$ of $E^*$ we are able to construct an infinite sequence of elements of $E^*$ that converges to $q$.
Rudin implicitly(!) assumes that $E^* subset X$, and as we are able to construct a sequence some points of $X$ taken from $E^*$, the limit $q$ of this subsequence belongs to $E^*$ by the way we chose to construct this sequence with elements of $E^*$.
Complete induction goes from the base case where we are picking points (each called $x$) starting from an arbitrary point $p_{n1} in E^* : p_{n1} ne q$ from smaller and smaller neighborhoods around $q$. The next one has radius that is half as small as the previous neighborhood of $q$. Since $q$ is a limit point, by Theorem 2.20, there are infinitely many points in $q$'s neighborhood that belong to $E^*$. So we are done and we are happy. $Box$
The point that concerned me about the proof is that Rudin did not restrict $X$ to be either compact, or $mathcal{R}$ in order to use Heine-Borel as a leverage here. He just made a ton of very important implicit assumptions and went along.
Suppose, we have $mathcal{Q} = X$ for this case. Since $mathcal{Q}$ is countable for this case and we can construct subsequences with irrational numbers as their limits. We can go along with Rudin's proof and get nothing.
The crucial point in this case is to play with the assumption that subsequential limits are part of $X$ (or not). This makes one think a lot over Rudin's explanations in this case. I understand that the empty set is closed, but Rudin's wording basically fails.
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
$endgroup$
add a comment |
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$begingroup$
The hard bit in this theorem to show that $E^* subset X$. In his proof Rudin is trying to make a direct proof by complete induction that for any limit point $q$ of $E^*$ we are able to construct an infinite sequence of elements of $E^*$ that converges to $q$.
Rudin implicitly(!) assumes that $E^* subset X$, and as we are able to construct a sequence some points of $X$ taken from $E^*$, the limit $q$ of this subsequence belongs to $E^*$ by the way we chose to construct this sequence with elements of $E^*$.
Complete induction goes from the base case where we are picking points (each called $x$) starting from an arbitrary point $p_{n1} in E^* : p_{n1} ne q$ from smaller and smaller neighborhoods around $q$. The next one has radius that is half as small as the previous neighborhood of $q$. Since $q$ is a limit point, by Theorem 2.20, there are infinitely many points in $q$'s neighborhood that belong to $E^*$. So we are done and we are happy. $Box$
The point that concerned me about the proof is that Rudin did not restrict $X$ to be either compact, or $mathcal{R}$ in order to use Heine-Borel as a leverage here. He just made a ton of very important implicit assumptions and went along.
Suppose, we have $mathcal{Q} = X$ for this case. Since $mathcal{Q}$ is countable for this case and we can construct subsequences with irrational numbers as their limits. We can go along with Rudin's proof and get nothing.
The crucial point in this case is to play with the assumption that subsequential limits are part of $X$ (or not). This makes one think a lot over Rudin's explanations in this case. I understand that the empty set is closed, but Rudin's wording basically fails.
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
$endgroup$
add a comment |
$begingroup$
The hard bit in this theorem to show that $E^* subset X$. In his proof Rudin is trying to make a direct proof by complete induction that for any limit point $q$ of $E^*$ we are able to construct an infinite sequence of elements of $E^*$ that converges to $q$.
Rudin implicitly(!) assumes that $E^* subset X$, and as we are able to construct a sequence some points of $X$ taken from $E^*$, the limit $q$ of this subsequence belongs to $E^*$ by the way we chose to construct this sequence with elements of $E^*$.
Complete induction goes from the base case where we are picking points (each called $x$) starting from an arbitrary point $p_{n1} in E^* : p_{n1} ne q$ from smaller and smaller neighborhoods around $q$. The next one has radius that is half as small as the previous neighborhood of $q$. Since $q$ is a limit point, by Theorem 2.20, there are infinitely many points in $q$'s neighborhood that belong to $E^*$. So we are done and we are happy. $Box$
The point that concerned me about the proof is that Rudin did not restrict $X$ to be either compact, or $mathcal{R}$ in order to use Heine-Borel as a leverage here. He just made a ton of very important implicit assumptions and went along.
Suppose, we have $mathcal{Q} = X$ for this case. Since $mathcal{Q}$ is countable for this case and we can construct subsequences with irrational numbers as their limits. We can go along with Rudin's proof and get nothing.
The crucial point in this case is to play with the assumption that subsequential limits are part of $X$ (or not). This makes one think a lot over Rudin's explanations in this case. I understand that the empty set is closed, but Rudin's wording basically fails.
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
$endgroup$
add a comment |
$begingroup$
The hard bit in this theorem to show that $E^* subset X$. In his proof Rudin is trying to make a direct proof by complete induction that for any limit point $q$ of $E^*$ we are able to construct an infinite sequence of elements of $E^*$ that converges to $q$.
Rudin implicitly(!) assumes that $E^* subset X$, and as we are able to construct a sequence some points of $X$ taken from $E^*$, the limit $q$ of this subsequence belongs to $E^*$ by the way we chose to construct this sequence with elements of $E^*$.
Complete induction goes from the base case where we are picking points (each called $x$) starting from an arbitrary point $p_{n1} in E^* : p_{n1} ne q$ from smaller and smaller neighborhoods around $q$. The next one has radius that is half as small as the previous neighborhood of $q$. Since $q$ is a limit point, by Theorem 2.20, there are infinitely many points in $q$'s neighborhood that belong to $E^*$. So we are done and we are happy. $Box$
The point that concerned me about the proof is that Rudin did not restrict $X$ to be either compact, or $mathcal{R}$ in order to use Heine-Borel as a leverage here. He just made a ton of very important implicit assumptions and went along.
Suppose, we have $mathcal{Q} = X$ for this case. Since $mathcal{Q}$ is countable for this case and we can construct subsequences with irrational numbers as their limits. We can go along with Rudin's proof and get nothing.
The crucial point in this case is to play with the assumption that subsequential limits are part of $X$ (or not). This makes one think a lot over Rudin's explanations in this case. I understand that the empty set is closed, but Rudin's wording basically fails.
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
$endgroup$
The hard bit in this theorem to show that $E^* subset X$. In his proof Rudin is trying to make a direct proof by complete induction that for any limit point $q$ of $E^*$ we are able to construct an infinite sequence of elements of $E^*$ that converges to $q$.
Rudin implicitly(!) assumes that $E^* subset X$, and as we are able to construct a sequence some points of $X$ taken from $E^*$, the limit $q$ of this subsequence belongs to $E^*$ by the way we chose to construct this sequence with elements of $E^*$.
Complete induction goes from the base case where we are picking points (each called $x$) starting from an arbitrary point $p_{n1} in E^* : p_{n1} ne q$ from smaller and smaller neighborhoods around $q$. The next one has radius that is half as small as the previous neighborhood of $q$. Since $q$ is a limit point, by Theorem 2.20, there are infinitely many points in $q$'s neighborhood that belong to $E^*$. So we are done and we are happy. $Box$
The point that concerned me about the proof is that Rudin did not restrict $X$ to be either compact, or $mathcal{R}$ in order to use Heine-Borel as a leverage here. He just made a ton of very important implicit assumptions and went along.
Suppose, we have $mathcal{Q} = X$ for this case. Since $mathcal{Q}$ is countable for this case and we can construct subsequences with irrational numbers as their limits. We can go along with Rudin's proof and get nothing.
The crucial point in this case is to play with the assumption that subsequential limits are part of $X$ (or not). This makes one think a lot over Rudin's explanations in this case. I understand that the empty set is closed, but Rudin's wording basically fails.
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
answered Jan 25 '17 at 22:19
Mikhail DMikhail D
36326
36326
add a comment |
add a comment |
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$begingroup$
This is a rather long restatement of Rudin's proof. I'm sure there are people (myself included) who will be happy to help but don't necessarily want to proofread the whole thing. Can you highlight the part(s) that you are not sure about?
$endgroup$
– Bungo
Apr 19 '16 at 19:08
1
$begingroup$
@Bungo please have a look at my post after I've edited it to remove some errors. Do you really think my rendering of Rudin's proof is too lengthy for other people to be able to take time reading it? Actually, this proof has been proving to be a bit too hard for me to imbibe for such a long time. So I thought I should try to write it out the way I've understood it and ask for the Maths SE community's opinion as to my understanding and presentation of the proof.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:15
2
$begingroup$
In general, a shorter more targeted question will attract more readers. I think including the full proof is OK, but it would be helpful if you can point out any part(s) that you would like us to focus on. Also, may I suggest adding the "proof-verification" tag.
$endgroup$
– Bungo
Apr 19 '16 at 19:23
$begingroup$
@Bungo what I'm specifically interested to know is whether or not I've read Rudin's inductive argument correctly, and whether or not I've managed to present it in a more easily understandable way.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:43
$begingroup$
@Bungo please have a look at my post now.
$endgroup$
– Saaqib Mahmood
Apr 19 '16 at 19:44